mathematics

Article

An Extended Finite Element Method (XFEM) Study on theElastic T-Stress Evaluations for a Notch in a Pipe Steel Exposedto Internal Pressure

K. Yakoubi 1, S. Montassir 1 , Hassane Moustabchir 2, A. Elkhalfi 1 , Catalin Iulian Pruncu 3,4,* , J. Arbaoui 5

and Muhammad Umar Farooq 6,7

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Citation: Yakoubi, K.; Montassir, S.;

Moustabchir, H.; Elkhalfi, A.; Pruncu,

C.I.; Arbaoui, J.; Farooq, M.U. An

Extended Finite Element Method

(XFEM) Study on the Elastic T-Stress

Evaluations for a Notch in a Pipe

Steel Exposed to Internal Pressure.

Mathematics 2021, 9, 507. https://

doi.org/10.3390/math9050507

Academic Editor: Krzysztof

Kamil Zur

Received: 23 January 2021

Accepted: 22 February 2021

Published: 2 March 2021

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4.0/).

1 Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Fez 30000, Morocco;khadija.yakoubi95@gmail.com (K.Y.); soufianemontassir@gmail.com (S.M.); aelkhalfi@gmail.com (A.E.)

2 Laboratory of Systems Engineering and Applications (LISA), National School of Applied Sciences of Fez,Sidi Mohamed Ben Abdellah University, Fez 30000, Morocco; hmoustabchir@hotmail.com

3 Department of Mechanical Engineering, Imperial College London, Exhibition Rd., London SW7 2AK, UK4 Design, Manufacturing & Engineering Management, University of Strathclyde, Glasgow G1 1XJ, UK5 National School of Applied Sciences of Safi, University Cadi Ayad, Marrakesh 40000, Morocco;

jamal57010@yahoo.fr6 Department of Industrial and Manufacturing Engineering, University of Engineering and Technology,

Lahore 54890, Pakistan; umarmuf0@gmail.com7 Department of Industrial and Systems Engineering, Advanced Institute of Science and Technology (KAIST),

Daejeon 34141, Korea* Correspondence: c.pruncu@imperial.ac.uk or Catalin.pruncu@strath.ac.uk

Abstract: The work investigates the importance of the K-T approach in the modelling of pressurecracked structures. T-stress is the constant in the second term of the Williams expression; it is oftennegligible, but recent literature has shown that there are cases where T-stress plays the role of openingthe crack, also T-stress improves elastic modeling at the point of crack. In this research study, the mostimportant effects of the T-stress are collected and analyzed. A numerical analysis was carried out bythe extended finite element method (X-FEM) to analyze T-stress in an arc with external notch underinternal pressure. The different stress method (SDM) is employed to calculate T-stress. Moreover, theinfluence of the geometry of the notch on the biaxiality is also examined. The biaxiality gave us aview on the initiation of the crack. The results are extended with a comparison to previous literatureto validate the promising investigations.

Keywords: T-stress; X-FEM; notch; pipe; stress difference method (SDM)

1. Introduction

In the field of fracture mechanics especially linear elastics, the vicinity of notch tip isoften symbolized by singular stress entities. Their resistance is measured through stressintensity factor (SIF). The SIF is mainly depended on the distance r from the tip of notch.In this field, parameter T is introduced to enrich the parameter K (SIF) to make the modelbetter in the elastic stress field; this is the K-T approach [1].

In fracture mechanics body of knowledge, it is established that the same Stress Inten-sity Factor (SIF) is required for two cracks to propagate in the same way. Experiment [2]have shown that two plates with the same SIF and different crack length a1 > a2 showdifferent the propagation of the cracks. The results have shown that the propagationspeed of a2 is higher than that of a1. The study concluded that the first term of asymptoticdevelopment is not sufficient to predict crack behavior. Therefore, it is necessary to increasethe order. The first term asymptotic development is the SIF that determines the initiationand propagation of the crack. In addition, the second term is constant and controls thestability of the propagation direction. It is the transverse component symbolized by T.

Mathematics 2021, 9, 507. https://doi.org/10.3390/math9050507 https://www.mdpi.com/journal/mathematics

Mathematics 2021, 9, 507 2 of 10

Several studies have shown the significance of the T-Stress, and its influence on differ-ent parameters of mechanics. Jayadevan [3] highlighted that plastic zone is manipulatedby the variation of T-stress. It means, the plastic area escalates with the increase in absolutevalue of T-stress and changes its form. Sobotka et al. [4] demonstrated the alteration in theplastic wake with many T-stresses which depends on plastic wakes height (HPW).

T-Stress has been an essential contributing factor in the stability of the direction of thecrack propagation. For instance, T negative gives a stable direction, and for T positive it isunstable [5]. Fayed et al. [6] explored the impact of T-stress on propagating crack directionby Maximum Tangential Stress (MTS). The principle of MTS is that the crack propagates inthe trend of maximum tangential stress. They obtained that the directions of the overallcrack no coincide with the initial direction of the crack. Many studies have concluded thebehavior by the fact that the tangential stress is affected by the T-Stress. MTS becomesgeneralized maximum tangential stress (GMTS), which considers the constraint T in theexpression of stress. In the same context Shahani [1] studied the result on the initiatingangle of propagating crack of the stress T. The study has shown a negative T value declinesthe angle of crack initiation, and a positive T value enhances it. Nejati et al. [7] gaugedthe relationship between T-stress and material properties. Chen et al. [8] has shown thatGraded Poisson’s ratio affects the T-stress. Additionally, Toshio et al. [9] concluded that thePoison’s ratio influences the T-stress on a three-dimensional edge-cracked plate.

Other important research that has shown the influences of the T-stress includes:Zhang et al. [10], which used numerical manifold method (NMM) is employed to calculatethe T-stress for two-dimensional functionally graded material (FGM) having numerouscracks. Noritaka et al. [11] has resulted the T-stress might open micro-branches in the mistregion. For the bending and tension load, Hancock [12] determined that T decreases withescalating crack length. Matvienko [13] explored the influence of T-stress in problems ofthe elastic and the elastic-plastic fracture mechanics.

Conventionally, the T-stress is often calculated at the crack tip, which is certainly notthe case in this research. The research evaluates the T-stress at the tip of notch throughextended finite element method (X-FEM).

The finite element method FE method is limited by the simple cases, as well as thepresence of a singularity greatly degrades the convergence of the FEM. Belytschko andBlacken in 1999 added discontinuously enriching function in finite element approximationby respecting boundary conditions. Later, Moës et al. [14] developed the technique andcalled it as extended finite element method which is abbreviated as X-FEM. The efficiencyof the X-FEM is well endorsed in the literature. To simulate the propagation of cracks inporous media, Wang et al. [15] integrated embedded discrete fracture method (EDFM) withX-FEM simulating fracture associated fluid and solid mechanics. Shu et al. [16] investigatedthe fatigue growth of 3-D multiple cracks by X-FEM. The implementation of X-FEM forcomposites resulted successfully [17–19]. X-FEM is also used for the calculation andanalysis of failure mechanics parameters. Fakkoussi et al. [20] calculated stress intensityfactor for mode one by X-FEM. Llavori et al. [21] studied the problems of contact fatigueby X-FEM.

The research study presents the use of X-FEM to calculate T-stress in the notch tip foran arc of the pipe of steel P264GH. Further, it demonstrates the benefits of using the X-FEMapproach to compute K-T at the notch point in an arc under pression and revealing thepossibility of detecting the crack initiation.

The remaining article is organized as, Section 2 talks about the K-T approach, X-FEM,and the geometry used. Section 3 deals with the numerical result obtained, comparedagainst FEM conclusions, along with the discussion. Finally, the conclusion is in theother section.

Mathematics 2021, 9, 507 3 of 10

2. Materials and Methods2.1. K-T Approach

Stress Intensity Factor has been an essential parameter in the field of linear fracturemechanics, widely used for crack evaluation. SIF measures the strength of the singularity,and integrates various parameters such as load, geometry, and shape of crack. M. HadjMeliani [22] suggested that a thin structure such as a thin pipe with a longitudinal crack, itis very difficult to characterize the stress field by a single parameter. For linear elasticity,the enrichment of the SIF with the T-stress is required to model the notch tip. In literature,studies show that T explains how geometry influences tenacity (KIC) [22]. T-stress helpsin approximating the level of stress at a crack or tip of the notch. The possibility ofconstructing the K(T) curve numerically has given the opportunity to predict the loadingof a crack initiation [23,24]. Including T-stress in calculations, it improves the prediction ofpropagating crack under the control of mixed loading. The K-T was additionally calculatedfor through-wall-cracked pipes under various pressure conditions by three dimension-3DFE [25]. The importance of the T-Stress is established in many works.

• T-stress enhances the possibility of crack opening stresses in the context of smallcrack [26];

σxx(r, θ) =K1√2πr

fxx(θ) + T (1)

Taking σxx(r, θ) = σcr, for the crack propagation, the first term tends to K1√2πr

fxx(θ) to

a because K1 = Syy√πa, and if a tends to zero; the first term becomes negligible intheface

to T at the crack point.lima→0

σxx(r, θ) = σcr = T (2)

In this case, the T-stress cannot be ignored, T play the role of crack opening, andtherefore the importance of T varies with the size of the cracks.

• T-stress influences the plastic zone. Jayadevan [3] highlighted that plastic zone isaffected by the variation of T-stress. The plastic area escalates with the upsurge in theabsolute value of T-stress and changes its form. Sobotka et al. [4] demonstrated thedeviation in the plastic wake with various T-stress which depends on plastic wakesheight (hpw).

• Propagation’s direction: Fayed et al. [6] analyzed the impact of T-stress on the prop-agation’s direction by Maximum Tangential Stress (MTS). The principle of MTS isin the crack propagation direction which is in line with that of maximum tangentialstress. MTS only considers the term singularity. Therefore, the direction of crackdoes not coincide with the initial directions of the crack. This behavior shows thatthe tangential stress is affected by the T-Stress. MTS becomes generalized maximumtangential stress (GMTS), which considers the T-stress in the expression of stress.

• T-stress has an impact on crack initiation angle. Shahani [1] analyzed the consequenceof the T-stress on the angle of initiation of crack propagation. The study has shownthat a negative T value declines the angle of crack initiation, and a positive T valueenhances it.

• T-Stress is of prime significance when ensuring the stability of the direction of crackpropagation, such as T negative gives a stable direction, and for T positive it isunstable [5].

T-Stress could be computed through numerous techniques. Weight Function Methodhas shown its efficiency in several problems cracking-related such as edge-cracked rect-angular plate, circular disk [27]. Kfouri [28] developed a technique for evaluating theT-stress. The method uses the attributes of the path-independent J-integral and is calledthe Esheby–Integral method.

Mathematics 2021, 9, 507 4 of 10

The stress different method (SDM) has been proposed by Yang [29]. The idea of thismethod is the errors of the numerical values of σ11 and σ22 near a crack point progresswith r in the same way, and the variation must effectively eliminate errors.

T = σyy − σxx (3)

Biaxiality is a parameter that relates the SIF and T-stress:

β =T√πa

K(4)

2.2. Extended Finite Elements

One of the uses of the FE method is the study of crack propagation but is limitedfor simple cases. If the mesh size does not conform to the crack, the FE method does nottreat the propagation, and the presence of a singularity degrades the convergence of theFE method.

The solution is to add enrichment function to the FE approximation (see Figure 1);this is the extended finite element method.

U =N

∑1

Niui +Nsaut

∑i

NiH(x)ai +

Nsin g

∑i

∑j

N(x)iF(x)jbji (5)

where:

• H(x): The Heaviside enrichment function, H(x) ={−1, x > 0+1, x < 0

• F(x): Enrichment functions near the crack front.• N: Interpolation function of finite element. Nsaut: Number of nodes enriched with

Heaviside function. Nsin g: Number of knots enriched near the crack front.

Mathematics 2021, 9, x FOR PEER REVIEW 4 of 11

edge-cracked rectangular plate, circular disk [27]. Kfouri [28] developed a technique for

evaluating the T-stress. The method uses the attributes of the path-independent J-integral

and is called the Esheby–Integral method.

The stress different method (SDM) has been proposed by Yang [29]. The idea of this

method is the errors of the numerical values of σ11 and σ22 near a crack point progress

with r in the same way, and the variation must effectively eliminate errors. T = σyy − σxx

(3)

Biaxiality is a parameter that relates the SIF and T-stress:

β =T√πa

K (4)

2.2. Extended Finite Elements

One of the uses of the FE method is the study of crack propagation but is limited for

simple cases. If the mesh size does not conform to the crack, the FE method does not treat

the propagation, and the presence of a singularity degrades the convergence of the FE

method.

The solution is to add enrichment function to the FE approximation (see Figure 1);

this is the extended finite element method.

U = ∑ Niui

N

1

+ ∑ NiH(x)ai +

Nsaut

i

∑ ∑ N(x)iF(x)jbij

j

Nsing

i

(5)

where:

• H(x): The Heaviside enrichment function, H(x) = {−1, x > 0+1, x < 0

• F(x): Enrichment functions near the crack front.

• N: Interpolation function of finite element. Nsaut: Number of nodes enriched with

Heaviside function. Nsing: Number of knots enriched near the crack front.

Figure 1. Step of enrichment methods.

The X-FEM requires operation to confirm the enrichment status of a knot according

to its position in reference to the crack and to evaluate the functions H(x) and F(x). The

position (r, θ) in relation to the notch point is calculated herein to know if x is above or

below the crack. These operations are carried out using the level sets method. The tech-

nique for describing crack is known as level set method. In the X-FEM, it determines the

location of the crack and the crack point, and the position to apply discontinuous en-

richment and enrichment to the crack point figure1. Most importantly, the level set pro-

vides an instant result that helps track crack propagation, i.e., as the crack propagates,

enrichment at the crack front becomes discontinuous enrichment, and nodes (not en-

riched) become enriched. There are two-level functions, and the first describes the crack

surface (φ), second gives the crack front (ψ) [30].

The X-FEM method was applied in several studies. Yousheng Xie et al. [31] have

implemented the X-FEM method in the study of propagating crack in mixed mode, and

Figure 1. Step of enrichment methods.

The X-FEM requires operation to confirm the enrichment status of a knot according toits position in reference to the crack and to evaluate the functions H(x) and F(x). The posi-tion (r, θ) in relation to the notch point is calculated herein to know if x is above or belowthe crack. These operations are carried out using the level sets method. The technique fordescribing crack is known as level set method. In the X-FEM, it determines the locationof the crack and the crack point, and the position to apply discontinuous enrichment andenrichment to the crack point Figure 1. Most importantly, the level set provides an instantresult that helps track crack propagation, i.e., as the crack propagates, enrichment at thecrack front becomes discontinuous enrichment, and nodes (not enriched) become enriched.There are two-level functions, and the first describes the crack surface (ϕ), second gives thecrack front (ψ) [30].

The X-FEM method was applied in several studies. Yousheng Xie et al. [31] haveimplemented the X-FEM method in the study of propagating crack in mixed mode, andevaluated the crack initiation angle. Reference [32] has shown the performance of themethod. The study applied X-FEM to calculate SIF for 3D crack propagation problems

Mathematics 2021, 9, 507 5 of 10

for a Compact Tension C-T specimen. X-FEM was also implemented in the analysis ofbi-material interfaces, calculating service life and fatigue resistance [33]. An integrationbetween the X-FEM and embedded discrete fracture method (EDFM) is established forsimulation of the process of fluid fracture propagation in porous media [15]. Savenkovet al. [34] employed the X-FEM to represent the central surface of the crack. The applicationof X-FEM for composite models is also carried out supporting current investigation [35,36].

X-FEM was used to predict components failure from a different form of notch [37].Patria et al. [38] adopted X-FEM to study the mechanical attributes and fracture behavior of(Reinforced Polymeric Composites) RPC materials with single edge notch three-point bending.

2.3. Geometry

An arc of pipe containing a notch under pression was numerically analyzed usingX-FEM in ABAQUS software. The material used is a steel P264GH. The arc characterizedby an inner radius Ri = 219.55 mm and thickness t = 6.1 mm. More details on geometry, theshape of the notch, and boundary requirements used are illustrated in Figure 2 and Table 1.

Mathematics 2021, 9, x FOR PEER REVIEW 5 of 11

evaluated the crack initiation angle. Reference [32] has shown the performance of the

method. The study applied X-FEM to calculate SIF for 3D crack propagation problems for

a Compact Tension C-T specimen. X-FEM was also implemented in the analysis of

bi-material interfaces, calculating service life and fatigue resistance [33]. An integration

between the X-FEM and embedded discrete fracture method (EDFM) is established for

simulation of the process of fluid fracture propagation in porous media [15]. Savenkov et

al. [34] employed the X-FEM to represent the central surface of the crack. The application

of X-FEM for composite models is also carried out supporting current investigation

[35,36].

X-FEM was used to predict components failure from a different form of notch [37].

Patria et al. [38] adopted X-FEM to study the mechanical attributes and fracture behavior

of (Reinforced Polymeric Composites) RPC materials with single edge notch three-point

bending.

2.3. Geometry

An arc of pipe containing a notch under pression was numerically analyzed using

X-FEM in ABAQUS software. The material used is a steel P264GH. The arc characterized

by an inner radius Ri = 219.55 mm and thickness t = 6.1 mm. More details on geometry,

the shape of the notch, and boundary requirements used are illustrated in Figure 2 and

Table 1.

Table 1. Geometry properties and load.

Ri [mm] P [MPa] φ [deg] a [mm] t [mm] ρ [mm]

213.45 15 45 1.22–4.88 6.1 0.15

Figure 2. Details of the geometry and notch study, with boundary conditions.

The mechanical attributes of the material used are presented in Table 2, and the

chemical composition of the material are included in Table 3.

Table 2. Mechanical characteristics of P264GH.

Young’s Modulus Poisson’s Ratio Yield Stress Elongation to Fracture

207,000 MPa 0.3 430 MPa 35%

Table 3. Chemical composition of P264GH.

Material C Mn S Si P Al

Tested 0.135 0.665 0.002 0.195 0.027 0.027

Steel P264GH

(Standard

EN10028.2–92)

0.18 1 0.015 0.4 0.025 0.02

Figure 2. Details of the geometry and notch study, with boundary conditions.

Table 1. Geometry properties and load.

Ri [mm] P [MPa] ϕ [deg] a [mm] t [mm] ρ [mm]

213.45 15 45 1.22–4.88 6.1 0.15

The mechanical attributes of the material used are presented in Table 2, and thechemical composition of the material are included in Table 3.

Table 2. Mechanical characteristics of P264GH.

Young’s Modulus Poisson’s Ratio Yield Stress Elongation toFracture

207,000 MPa 0.3 430 MPa 35%

Table 3. Chemical composition of P264GH.

Material C Mn S Si P Al

Tested 0.135 0.665 0.002 0.195 0.027 0.027

Steel P264GH(Standard EN10028.2–92) 0.18 1 0.015 0.4 0.025 0.02

3. Results

This section presented the results of SIF and T-stress given by X-FEM via an userelement UEL subroutine, the calculation was executed by ABAQUS software, we used

Mathematics 2021, 9, 507 6 of 10

quadratic element C3D20 in the mesh Figure 3, with a size of 0.5 mm. The number ofelements is 246117. The number of nodes is 363256.

Mathematics 2021, 9, x FOR PEER REVIEW 6 of 11

3. Results

This section presented the results of SIF and T-stress given by X-FEM via an

user element UEL subroutine, the calculation was executed by ABAQUS soft-

ware, we used quadratic element C3D20 in the mesh Figure 3, with a size of 0.5

mm. The number of elements is 246117. The number of nodes is 363256.

Figure 3. The mesh of 3D arc, C3D10 with 0.5 mm of size.

The T-stress is calculated through stress different method, and normaliza-

tion is done for effect of the T-stress relative to the stress intensity factor by a

parameter dimensionless termed biaxiality.

𝛽 =T√πa

𝐾

K is the value of stress intensity factor, and a is the notch length.

Figure 4 illustrates the difference of the SIF in mode 1 as a function of r, for

a/t = 0.2, by extended finite element.

Figure 4. Distribution stress intensity factor SIF at the notch tip–SIF and r for a/t = 0.2.

The elastic SIF distribution at the notch tip decreases with distance from the

tip of notch, for a/t = 0.2 the maximum value of SIF is 21 MPa√m at notch tip, i.e.,

𝑟 = 0 (see Figure 4). Near the notch tip SIF decreases rapidly to 6 MPa√m at r =

0.3 mm, then its variation becomes slower.

Figure 5 shows the variation of stress σxx,σyy and T. T-stress increases with

increasing r up to r = 0.43, and after that it starts to stabilize. The numerical cal-

0.0 0.2 0.4 0.6 0.8 1.0

2

4

6

8

10

12

14

16

18

20

22

Str

ess

Inte

nsi

ty F

acto

r K

I [M

Pa

m0

.5]

r [mm]

K I Numerical (XFEM)

Figure 3. The mesh of 3D arc, C3D10 with 0.5 mm of size.

The T-stress is calculated through stress different method, and normalization is donefor effect of the T-stress relative to the stress intensity factor by a parameter dimensionlesstermed biaxiality.

β =T√πa

KK is the value of stress intensity factor, and a is the notch length.Figure 4 illustrates the difference of the SIF in mode 1 as a function of r, for a/t = 0.2,

by extended finite element.

Mathematics 2021, 9, x FOR PEER REVIEW 6 of 11

3. Results

This section presented the results of SIF and T-stress given by X-FEM via an

user element UEL subroutine, the calculation was executed by ABAQUS soft-

ware, we used quadratic element C3D20 in the mesh Figure 3, with a size of 0.5

mm. The number of elements is 246117. The number of nodes is 363256.

Figure 3. The mesh of 3D arc, C3D10 with 0.5 mm of size.

The T-stress is calculated through stress different method, and normaliza-

tion is done for effect of the T-stress relative to the stress intensity factor by a

parameter dimensionless termed biaxiality.

𝛽 =T√πa

𝐾

K is the value of stress intensity factor, and a is the notch length.

Figure 4 illustrates the difference of the SIF in mode 1 as a function of r, for

a/t = 0.2, by extended finite element.

Figure 4. Distribution stress intensity factor SIF at the notch tip–SIF and r for a/t = 0.2.

The elastic SIF distribution at the notch tip decreases with distance from the

tip of notch, for a/t = 0.2 the maximum value of SIF is 21 MPa√m at notch tip, i.e.,

𝑟 = 0 (see Figure 4). Near the notch tip SIF decreases rapidly to 6 MPa√m at r =

0.3 mm, then its variation becomes slower.

Figure 5 shows the variation of stress σxx,σyy and T. T-stress increases with

increasing r up to r = 0.43, and after that it starts to stabilize. The numerical cal-

0.0 0.2 0.4 0.6 0.8 1.0

2

4

6

8

10

12

14

16

18

20

22

Str

ess

Inte

nsi

ty F

acto

r K

I [M

Pa

m0

.5]

r [mm]

K I Numerical (XFEM)

Figure 4. Distribution stress intensity factor SIF at the notch tip–SIF and r for a/t = 0.2.

The elastic SIF distribution at the notch tip decreases with distance from the tip ofnotch, for a/t = 0.2 the maximum value of SIF is 21 MPa

√m at notch tip, i.e., r = 0

(see Figure 4). Near the notch tip SIF decreases rapidly to 6 MPa√

m at r = 0.3 mm, then itsvariation becomes slower.

Figure 5 shows the variation of stress σxx,σyy and T. T-stress increases with increasingr up to r = 0.43, and after that it starts to stabilize. The numerical calculation of the stressσxx,σyy by X-FEM is executed by ABAQUS software. Figure 6 gives the Von Mises stressobtained by the Abaqus software.

Mathematics 2021, 9, 507 7 of 10

Mathematics 2021, 9, x FOR PEER REVIEW 7 of 11

culation of the stress σxx,σyy by X-FEM is executed by ABAQUS software. Figure

6 gives the Von Mises stress obtained by the Abaqus software.

Figure 5. T-stress by stress different method.

Figure 6. Distributions of Von Mises stress near the notch.

The result obtained of biaxiality are compared with H. Moustabchir [39],

who calculated the biaxiality through the finite element method, for the same

geometry which is used in this study. Figure 7 gives the variation of biaxiality as

a function of a/t, by X-FEM and FEM.

The biaxiality levels up with the increase a/t, The results are identical with the H.

Moustabchir [39] recommendations. Kim and Paulino [40] has also got the same varia-

tion. By X-FEM the biaxiality varies from −0.21 to 0.76 for a/t = 0.2 and 0.8, respectively,

and it goes from a positive to a negative value at a/t = 0.3. The difference between the

results given by X-FEM and FEM is 0.013.

0.0 0.2 0.4 0.6 0.8 1.0-300

-200

-100

0

100

200

300

400

Str

ess

[MP

a]

r [mm]

σxx

σyy

T-Stress

Figure 5. T-stress by stress different method.

Mathematics 2021, 9, x FOR PEER REVIEW 7 of 11

culation of the stress σxx,σyy by X-FEM is executed by ABAQUS software. Figure

6 gives the Von Mises stress obtained by the Abaqus software.

Figure 5. T-stress by stress different method.

Figure 6. Distributions of Von Mises stress near the notch.

The result obtained of biaxiality are compared with H. Moustabchir [39],

who calculated the biaxiality through the finite element method, for the same

geometry which is used in this study. Figure 7 gives the variation of biaxiality as

a function of a/t, by X-FEM and FEM.

The biaxiality levels up with the increase a/t, The results are identical with the H.

Moustabchir [39] recommendations. Kim and Paulino [40] has also got the same varia-

tion. By X-FEM the biaxiality varies from −0.21 to 0.76 for a/t = 0.2 and 0.8, respectively,

and it goes from a positive to a negative value at a/t = 0.3. The difference between the

results given by X-FEM and FEM is 0.013.

0.0 0.2 0.4 0.6 0.8 1.0-300

-200

-100

0

100

200

300

400

Str

ess

[MP

a]

r [mm]

σxx

σyy

T-Stress

Figure 6. Distributions of Von Mises stress near the notch.

The result obtained of biaxiality are compared with H. Moustabchir [39], who calcu-lated the biaxiality through the finite element method, for the same geometry which isused in this study. Figure 7 gives the variation of biaxiality as a function of a/t, by X-FEMand FEM.

Mathematics 2021, 9, x FOR PEER REVIEW 8 of 11

Figure 7. Variation of the biaxiality with a/t at the notch.

4. Discussion

SIF measures the strength of the singularity, which explains the rapid variation of

the SIF obtained for r = [0 , 0.3]. The more approach is made towards the notch tip, the

more the stress concentration increases, and therefore, SIF increases. Moustabchir et al.

[39] used the volumetric approach to calculate SIF, in the same condition that this re-

search studied. Moustabchir et al. obtained for mode I at tip notch K1 = 21.6 MPa√m,

which differs from our result by 0.5. The T-stress can be analyzed by the SDM. The biax-

iality increases with the rise in a/t, and the notch depth affects the value of T-stress. The

same variation was obtained in other investigations on various materials. Bouchard et al.

[41] have shown that T increases with increasing depth for a mono silicon. In [42], Sherry

et al. obtained an increase in T in absolute value with the variation of the crack size over

the width of a plate. In addition, Ayatollahi et al. [43] obtained for mode I, an increase in

T-stress as a function of the depth of the crack for a single edge notched.

If

β = 1

and

β =T√πa

K

So

K = T√πa i.e., T = σ

Which is not the case for this study 𝛽𝑚𝑎𝑥 = 0.78, so T ≠ σ, which implies that T has

no influence on the notch in our case and our condition.

Many studies have resulted that the influence of the T-stress is remarkable and sig-

nificant when T is negative [1,3,6], however, in this study for r < 0.43, negative T-stress

causes an increase in the plastic zone [27]. This will cause a crack to initiate. In the pres-

ence of a crack, negative T can change the direction of propagating crack and decreases

the crack growth initiation angle [31].

The K-T approach is an integration between the SIF and the T-stress to improve

modelling the elastic stress at the point of the crack. The importance of T-stress has been

highlighted, and it takes the place of short crack opening stress. Besides, the importance

of the cooperation of SIF and T-stress, such as the K(T) curve was elaborated which

0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Bia

xia

lity

b

a/t

Present study (XFEM)

H.MOUSTABCHIR

KIM and PAILIMO

Figure 7. Variation of the biaxiality with a/t at the notch.

Mathematics 2021, 9, 507 8 of 10

The biaxiality levels up with the increase a/t, The results are identical with the H.Moustabchir [39] recommendations. Kim and Paulino [40] has also got the same variation.By X-FEM the biaxiality varies from −0.21 to 0.76 for a/t = 0.2 and 0.8, respectively, and itgoes from a positive to a negative value at a/t = 0.3. The difference between the resultsgiven by X-FEM and FEM is 0.013.

4. Discussion

SIF measures the strength of the singularity, which explains the rapid variation of theSIF obtained for r = [0, 0.3]. The more approach is made towards the notch tip, the morethe stress concentration increases, and therefore, SIF increases. Moustabchir et al. [39] usedthe volumetric approach to calculate SIF, in the same condition that this research studied.Moustabchir et al. obtained for mode I at tip notch K1 = 21.6 MPa

√m, which differs from

our result by 0.5. The T-stress can be analyzed by the SDM. The biaxiality increases withthe rise in a/t, and the notch depth affects the value of T-stress. The same variation wasobtained in other investigations on various materials. Bouchard et al. [41] have shownthat T increases with increasing depth for a mono silicon. In [42], Sherry et al. obtained anincrease in T in absolute value with the variation of the crack size over the width of a plate.In addition, Ayatollahi et al. [43] obtained for mode I, an increase in T-stress as a functionof the depth of the crack for a single edge notched.

Ifβ = 1

and

β =T√πa

KSo

K = T√πa i.e., T = σ

Which is not the case for this study βmax = 0.78, so T 6= σ, which implies that T hasno influence on the notch in our case and our condition.

Many studies have resulted that the influence of the T-stress is remarkable and signifi-cant when T is negative [1,3,6], however, in this study for r < 0.43, negative T-stress causesan increase in the plastic zone [27]. This will cause a crack to initiate. In the presence of acrack, negative T can change the direction of propagating crack and decreases the crackgrowth initiation angle [31].

The K-T approach is an integration between the SIF and the T-stress to improvemodelling the elastic stress at the point of the crack. The importance of T-stress has beenhighlighted, and it takes the place of short crack opening stress. Besides, the importance ofthe cooperation of SIF and T-stress, such as the K(T) curve was elaborated which gives aprediction of the stress of crack initiation [44]. Neggaz et al. [44] studied the influences ofthe reinforcements in the structure of composites, with the aim of reducing constraints atnotch-tip. Moreover, authors evaluated the effective stress intensity factors in the regard ofpropagating crack in thin and thick panels. Therefore, an Extended Finite Element Method(XFEM) is novel and improved technique on the elastic T-stress evaluations for a notch in apipe steel exposed to internal pressure.

5. Conclusions

Three-dimensional Extended Finite Element (X-FEM) analysis is applied to evaluatethe stress intensity factor and the T-stress for an arc of pipe with external notch underinternal pressure. The results are presented below:

• To study the influence of geometry and notch size on the T-stress, authors haveapproached the biaxiality as a function of a/t. The evaluation endorsed that biaxialityβ increases with the increasing a/t which is in accord with president results.

• The integration of biaxiality allowed us to determine the state of the crack initiation,and we can say that the pipe is safe in the used conditions.

Mathematics 2021, 9, 507 9 of 10

• The SIF alone does not characterize the behavior of notches. T-stress is obtained byStress Difference Method (SDM) along with the notch in mode I. SDM is an efficientand simple method to calculate the fracture parameters.

• With ABAQUS-based investigations, the numerical results achieved by X-FEM are ingood agreement with the Moustabchir result [39]. The implementation of X-FEM inthe presence of a notch corrected the problems of the standard finite element method.The advantage of the X-FEM is that the mesh is independent of the notch.

For more precision, the future objective is to calculate the parameters of fracturemechanics by iso-geometrical analysis.

Author Contributions: Data curation, K.Y. and J.A., formal analysis, S.M. and A.E., investigation,H.M. and C.I.P., writing and review of original draft, M.U.F., H.M., and C.I.P. All authors have readand agreed to the published version of the manuscript.

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: The data that support the findings of this study are available on requestfrom the corresponding author.

Acknowledgments: Authors are thankful to Aqib Mashood Khan for his constructive feedback.

Conflicts of Interest: The authors declare no conflict of interest.

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